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Solutions for Chapter 7.1: Modeling with Differential Equations

Biocalculus: Calculus for Life Sciences | 1st Edition | ISBN: 9781133109631 | Authors: James Stewart, Troy Day

Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition

ISBN: 9781133109631

Biocalculus: Calculus for Life Sciences | 1st Edition | ISBN: 9781133109631 | Authors: James Stewart, Troy Day

Solutions for Chapter 7.1: Modeling with Differential Equations

Solutions for Chapter 7.1
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Textbook: Biocalculus: Calculus for Life Sciences
Edition: 1
Author: James Stewart, Troy Day
ISBN: 9781133109631

This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 7.1: Modeling with Differential Equations have been answered, more than 26265 students have viewed full step-by-step solutions from this chapter. Chapter 7.1: Modeling with Differential Equations includes 18 full step-by-step solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1.

Key Math Terms and definitions covered in this textbook
  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

    Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Indefinite matrix.

    A symmetric matrix with eigenvalues of both signs (+ and - ).

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • lA-II = l/lAI and IATI = IAI.

    The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Unitary matrix UH = U T = U-I.

    Orthonormal columns (complex analog of Q).

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